Stability robustness of interval matrices via Lyapunov quadraticforms

A sufficient condition for the robust stability of a class of interval matrices is derived using the Lyapunov approach. The matrices considered have elements which are nonlinear functions of a vector of independent and bounded parameters. The robust stability condition requires that a quadratic form be positive definite in a finite number of conspicuous points of an enlarged parameter space     

The majorant Lyapunov equation: A nonnegative matrix equation for robust stability and performance of large scale systems

A new robust stability and performance analysis technique is developed. The approach involves replacing the state covariance by its block-norm matrix, i.e., the nonnegative matrix whose elements are the norms of subblocks of the covariance matrix partitioned according to subsystem dynamics. A bound (i.e., majorant) for the block-norm matrix is given by the majorant Lyapunov equation, a Lyapunov-type nonnegative matrix equation. Existence, uniqueness, and computational tractability of solutions to the majorant Lyapunov equation are shown to be completely characterized in terms ofMmatrices. Two examples are considered. For a damped simple harmonic oscillator with uncertain but constant natural frequency, the majorant Lyapunov equation predicts unconditional stability. And, for a pair of nominally uncoupled oscillators with uncertain coupling, the majorant Lyapunov equation shows that the range of nondestabilizing couplings is proportional to the frequency separation between the oscillators, a result not predictable from quadratic or vector Lyapunov functions.     

On the discrete Lyapunov matrix equation

Some bounds for the arithmetic and the geometric means of the characteristic roots of the positive semidefinite solution to the discrete Lyapunov matrix equation are derived.     

On the Lyapunov matrix differential equation

A lower bound for the determinant of the solution to the Lyapunov matrix differential equation is derived. It is shown that this bound is obtained as a solution to a simple scalar differential equation. In the limiting case where the solution to the Lyapunov differential equation becomes stationary, the result reduces to one of the existing bounds for the algebraic equation.     

A theorem on the Lyapunov matrix equation

Given the Lyapunov matrix equationA'P + PA + 2sigmaQ = 0where σ is some positive scalar, a necessary and sufficient condition for the real parts of the eigenvalues ofAto be less than -σ is thatP - Qis negative definite. The condition provides an upper bound to the solution of the Lyapunov matrix equation and is useful in the design of minimum-time or minimum-cost linear control systems.     

A note on the Lyapunov matrix equation

In [5] bound for the determinant of the solution to the Lyapunov matrix equation was reported. This note gives an another bound for this value.     

Bounds in the Lyapunov matrix differential equation

Upper and lower bounds for the trace of the solution of the Lyapunov matrix differential equation are derived. It is shown that they are obtained as solutions to simple scalar differential equations. As a special case, the bounds for the stationary solution give ones for the solution to the Lyapunov algebraic equation.     

Generalized Lyapunov equation and factorization of matrix polynomials

This paper presents an algorithm for the construction of a solution of the generalized Lyapunov equation. It is proved that the polynomial matrix factorization relative to the imaginary axis may be reduced to the successive solution of Lyapunov equations, i.e. the factorization is reduced to the solution of a sequence of generalized Lyapunov equations, not to the solution of generalized Riccati equation.     

Lyapunov stability robustness measures for multivariable,continuous, time-invariant,linear systems

In this paper general robustness measure bounds are introduced for any multivariable, continuous, time-invariant, linear systems. Bounds are obtained for allowable non-linear time-varying perturbations such that the resulting system remains stable. Bounds are also derived for linear perturbations. The robustness measures and the related theorems are applied to optimal LQ state feedback, direct output feedback, and to generalized dynamic output feedback designs.     

On Lyapunov stability robustness bounds for linear discrete-time systems with structured time-varying uncertainty

In this paper we analyse the stability robustness of linear discrete-time systems which are described by a state-space model but are perturbed with structured time-varying uncertainty. We present new Lyapunov stability robustness bounds in which the freedom of the matrix Q is utilized more effectively than that used by Kolla et al. (1989) to obtain a larger bound of tolerable time-varying uncertainty, and the similarity transformation is employed more directly and usefully than that proposed by Kolla and Farison (1990) to reduce conservation. Further, the relationship between the matrix Q and the similarity transformation matrix M is given. Improvements are illustrated by an application of our proposed method to a macroeconomic system     

Second-order n-dimensional systems and the Lyapunov matrix equation

Equations analogous to the Lyapunov matrix equation are derived for second-ordern-dimensional systems. These are shown to be more readily solvable than the equivalent2n-dimensional Lyapunov matrix equation.     

Improved gradient-based neural networks for online solution of Lyapunov matrix equation

By adding different activation functions, a type of gradient-based neural networks is developed and presented for the online solution of Lyapunov matrix equation. Theoretical analysis shows that any monotonically-increasing odd activation function could be used for the construction of neural networks, and the improved neural models have the global convergence performance. For the convenience of hardware realization, the schematic circuit is given for the improved neural solvers. Computer simulation results further substantiate that the improved neural networks could solve the Lyapunov matrix equation with accuracy and effectiveness. Moreover, when using the power-sigmoid activation functions, the improved neural networks have superior convergence when compared to linear models.     

Necessary conditions for the exponential stability of time‐delay systems via the Lyapunov delay matrix

Exponential necessary stability conditions for linear systems with multiple delays are presented. The originality of these conditions is that, in analogy with the case of delay free systems, they depend on the Lyapunov matrix function of the delay system. They are validated by examples for which the analytic characterization of the stability region is known. Copyright © 2013 John Wiley & Sons, Ltd.     

Construction of square root factor for solution of the Lyapunov matrix equation

The Lyapunov matrix equation is considered in this paper, where the solution is a nonnegative definite matrix, i.e. a matrix admitting decomposition in square root factors. An algorithm for findings the square root factor without preliminary finding the solution itself is given.     

Comments on "On the Lyapunov matrix equation"

The Lyapunov matrix equationA'Q + QA = - Pis considered in the above paper, where two fundamental inequalities are derived which are satisfied by the extremal eigenvalues of the matricesQandPprovidedAis a stability matrix. Similar results are derived by an alternate more simple and straightforward approach using matrix norms.     

2-D stability test via 1-D stability robustness analysis

A new two-dimensional stability test is proposed, based on the stability robustness analysis of a relevant 1-D system family. A linear algebra type algorithm for numerical implementation of the test is also described. Two examples then follow, to illustrate the use of the algorithm and the feasibility of the proposed test.     

Computational experience with the solution of the matrix Lyapunov equation

This correspondence presents a comparative study of three methods for the numerical solution of the matrix Lyapunov equation. The test case is a 24th-order system with highly underdamped eigenvalues and a rather high degree of stiffness. The conclusions favor a method by Bartels and Stewart based on a reduction to Schur form of theAmatrix.     

Improved neural solution for the Lyapunov matrix equation based on gradient search

By using the hierarchical identification principle, based on the conventional gradient search, two neural subsystems are developed and investigated for the online solution of the well-known Lyapunov matrix equation. Theoretical analysis shows that, by using any monotonically-increasing odd activation function, the gradient-based neural networks (GNN) can solve the Lyapunov equation exactly and efficiently. Computer simulation results confirm that the solution of the presented GNN models could globally converge to the solution of the Lyapunov matrix equation. Moreover, when using the power-sigmoid activation functions, the GNN models have superior convergence when compared to linear models.     

Explicit solutions of the discrete-time Lyapunov matrix equation and Kalman-Yakubovich equations

A general solution for the nonsquare nonsymmetric Lyapunov matrix equation in a canonical form is presented. The solution is shown to be a Toeplitz matrix which may be calculated using the backwards Levinson algorithm This solution is then applied to the Kalman-Yakubovich equations to derive a method for generating strictly positive-real functions via the positive-real lemma. This latter result has an application in system identification.